A central problem in the theory of algorithms for data streams is to determine which functions on a stream can be approximated in sublinear, and especially sub-polynomial or polylogarithmic, space. Given a function g, we study the space complexity of approximating n i=1 g(|fi|), where f ∈ Z n is the frequency vector of a turnstile stream. This is a generalization of the well-known frequency moments problem, and previous results apply only when g is monotonic or has a special functional form. Our contribution is to give a condition such that, except for a narrow class of functions g, there is a space-efficient approximation algorithm for the sum if and only if g satisfies the condition. The functions g that we are able to characterize include all convex, concave, monotonic, polynomial, and trigonometric functions, among many others, and is the first such characterization for non-monotonic functions. Thus, for nearly all functions of one variable, we answer the open question from the celebrated paper of Alon, Matias and Szegedy (1996).
International audience
We state and prove new properties about Doeblin's ergodicity coefficient for finite Markov chains. We show that this coefficient satisfies a sub-multiplicative type inequality (analogous to the Markov-Dobrushin's ergodicity coefficient), and provide a novel but elementary proof of Doeblin's characterization of weak-ergodicity for non-homogeneous chains. Using Doeblin's coefficient, we illustrate how to approximate a homogeneous but possibly non-stationary Markov chain of duration $n$ by independent and short-lived realizations of an auxiliary chain of duration of order $\ln (n)$. This leads to approximations of occupancy distributions in homogeneous chains, which may be particularly useful when exact calculations via one-step methods or transfer matrices are impractical, and when asymptotic approximations may not be yet reliable. Our findings may find applications to pattern problems in Markovian and non-Markovian sequences that are treatable via embedding techniques.
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